Disc e e
Dynamics
in
Na u e
and
Socie y,
2002
VoL.
7
(1),
pp.
41-52
Taylo
&
F ancis
Taylo
&
F ancis
G oup
E ec
o
Pa ame e
Calcula ion
in
Di ec
Es ima ion
o
he
Lyapuno
Exponen
in
Sho
Time
Se ies
A.M.
L(3PEZ
JIMINEZ
a’*,
C.
CAMACHO
MARTI’NEZ
VARA
DE
REY"
and
A.R.
GARC[A
TORRES
b
aDepa men
o
Expe imen al
Psychology,
Uni e si y
o
Se ille,
A da.
Camilo
Jos
Cela
s/n.
41005
Se ille,
Spain;
bphysics
Semina ,
LE.S.
Los
Vi e os,
A da.
Bias
In an e,
s/n.
Se ille,
Spain
(Recei ed
21
Ap il
2001)
The
li e a u e
abou
non-linea
dynamics
o e s
a
ew
ecommenda ions,
which
some imes
a e
di e gen ,
abou
he
c i e ia
o
be
used
in
o de
o
selec
he
op imal
calculus
pa ame e s
in
he
es ima ion
o
Lyapuno
exponen s
by
di ec
me hods.
These
ew
ecommenda ions
a e
ci cumsc ibed
o
he
analysis
o
chao ic
sys ems.
We
ha e
ound
no
ecommenda ion
o
he
es ima ion
o
A
s a ing
om
he
ime
se ies
o
classic
sys ems.
The
eason
o
his
is
he
in e es
in
dis inguishing
a iabili y
due
o
a
chao ic
beha io
o
de e minis
dynamic
sys ems
o
a iabili y
caused
by
whi e
noise
o
linea
s ochas ic
p ocesses,
and
less
in
he
iden i ica ion
o
non-linea
e ms
om
he
analysis
o
ime
se ies.
In
his
s udy
we
ha e
cen e ed
in
he
dependence
o
he
Lyapuno
exponen ,
ob ained
by
means
o
di ec
es ima ion,
o
he
ini ial
dis ance
and
he
ime
e olu ion.
We
ha e
used
gene a ed
se ies
o
chao ic
sys ems
and
gene a ed
se ies
o
classic
sys ems
wi h
a ying
complexi y.
To
gene a e
he
se ies
we
ha e
used
he
logis ic
map.
Keywo ds:
Non-linea
dynamic;
A ac o ;
Chaos;
Lyapuno
exponen
INTRODUCTION
The
disco e y
o
chao ic
beha io
in
de e minis ic
dynamical
sys ems
has
changed
some
philosophical
aspec s
in
he
p e ailing
scien i ic
pa adigm
and
has
opened
new
pe spec i es
o
he
design
and
analysis
o
ime
se ies
(Ba ne
and
Choi,
1989;
Casdagli,
1991;
Casdagli
e
al.,
1991;
Saye s,
1991;
Be line ,
1992;
McCa ey
e
al.,
1992;
Nychka
e
al.,
1992;
Ge
and
Allen,
1993;
Takens,
1993).
In
he
1980s,
he
b eak h oughs
in
he
analysis
o
ime
se ies
based
on
he
Quali a i e
Theo y
o
Dynamical
Sys ems
ha e
yielded
a
se
o
indexes.
These,
in
heo y,
should
allow
us
o
de e mine
i
he
appa en ly
andom
ime
sequence
obse a ions
o
a
sys em
s a e,
can
o
canno
be
due
o
chao ic
beha io
gene a ed
by
a
sys em
o
nonlinea
de e minis ic
equa ions
(Ashley
e
al.,
1986;
B oomhead
and
King,
1986;
Ashley
and
Pa e son,
1989;
B own
e
al.,
1991;
G assbe ge
e
al.,
1991;
McCa ey
e
al.,
1992;
Aba banel
e
al.,
1993;
Palu
e
al.,
1993;
Takens,
1993).
As
a
sub-p oduc ,
i
is
possible
o
de e mine
he
numbe
o
a iables
which
his
se
o
unknown
equa ions
would
b ing
in o
play,
as
well
as
o
classi y
sys ems
in o
uni e sal
classes
(linea -non-linea ,
s ochas ic-de e minis ic)
and
ela e
he
changes
in
he
beha io
quan i ie s
wi h
changes
occu ed
in
he
dynamical
beha io
o
he
sys em
(bi u ca ions)
(Sugiha a
and
May,
1990;
Mon e o
and
Mo in,
1992).
Al hough
cha ac e izing
dynamical
sys ems
using
he
analysis
o
uni-dimensional
ime
se ies
has
been
a
me hod
widely
de eloped
since
he
1980s,
he e
a e
se e al
ques ions
ha
need
some
conside a ion,
a
leas
in
he
cases
when
he
indica o s
a e
ob ained
om
ime
se ies
esul ing
om
beha io al
in es iga ion.
In
his
ype
o
in es iga ion,
like
in
mos
si ua ions
in
eal
li e,
he
da a
combine
de e minis ic
dynamics
wi h
noise
o
di e en
na u e
and
magni ude;
in
addi ion
o
his,
in
Psychology
i
is
di icul
o
main ain
he
same
obse a ion
si ua ion
o
a
long
ime
and
his
leads
o
a
educ ion
in
he
leng h
o
he
se ies,
he e o e
he
eliabili y
o
such
indexes
can
be
ques ionable.
In
his
s udy,
we
ha e
ied
o
p o ide
answe s
o
he
ques ions
a ising
om
he
calcula ion
o
dominan
Lyapuno
exponen
using
di ec
me hods.
We
ha e
*Co esponding
au ho .
Tel.:
+34-954557812.
Fax:
+34-954551784.
E-mail:
[email p o ec ed]
ISSN
1026-0226
(C)
2002
Taylo
&
F ancis
L d
42
A.M.L.
JIMNEZ
e
al.
cen e ed
on
analyzing
he
s eng h
o
he
exponen
o
di e en
alues
o
e olu ion
imes
and
ini ial
dis ances
in
sho
ime
se ies.
LYAPUNOV
EXPONENTS:
DIRECT
ESTIMATION
The
dominan
Lyapuno
exponen
is
one
o
he
mos
widely
used
indica o s
o
desc ibe
he
quali a i e
beha io
in
a
dynamical
sys em
using
he
analysis
o
uni-
dimensional
ime
se ies.
To
de ine
wha
is
unde s ood
by
Lyapuno
exponen
(A)
we
s a
om
an
ini ial
condi ion
Yo,
o
a
disc e e
dynamical
sys em
and
we
conside
a
e y
close
poin ,
whe e
he
ini ial
dis ance
(do)
is
ex emely
small.
Le
d
be
he
dis ance
a e
i e a ions.
I
we
assume
ha
in
Eq.
(3)
when
do
d aws
o
0,
he
e m
wi hin
he
loga i hm
is
he
de i a i e
o
he
i e a e
o
e alua ed
in
y0(( )(y0)).
Applying
he
chain
ule
o
di e en ia ion,
he
de i a i e
o
can
be
w i en
as
a
p oduc
o
de i a i es
o
y)
e alua ed
a
he
successi e
ajec o y
poin s
yo,yl,y2,..,
and
so
on.
We
can
hen
de ine
Lyapuno
exponen
in
a
mo e
in ui i e
way
wi h
he
ollowing
1
A
lim-
In
-1
H (Yl)
=0
liml
(lnl ’(y0)l
+
lnl (y)l
+...
--,eo
+
lnl ’(y,-l)l)
Id,
Id01
exp
( A)
(1)
hen
A
is
wha
we
call
Lyapuno
exponen
(Packa d
e
al.,
1980;
Schus e ,
1984;
Mon e o
and
Mo an,
1992;
Nychka
e
al.,
1992;
Aba banel
e
al.,
1993;
Simmons,
1993;
S oga z,
1994;
Hilbo n,
1994;
Ma in
e
al.,
1995).
Tha
is,
he
a e age
exponen ial
a e
o
di e gence
o
con e gence
o
ajec o ies
which
a e
e y
close
in
phase
space
(Wol
e
al.,
1985;
DeSouza-Machado
e
al.,
1990;
Zeng
e
al.,
1991).
The
numbe
o
Lyapuno
exponen s
in
a
dynamical
equals
he
numbe
o
s a e
a iables
conside ed.
A
uni-
dimensional
sys em
is
cha ac e ized
by
one
single
exponen .
The
signs
o
Lyapuno
exponen s
p o ide
quali a i e
in o ma ion
abou
he
dynamics
o
a
sys em.
I
he
sign
is
posi i e,
his
is
an
indica ion
o
chaos.
I
i
is
nega i e,
he e
is
con e gence
be ween
close
ajec o ies
and
he e o e
classic
a ac o s
exis .
I
he
beha io
o
a
dynamical
sys em
ep esen ed
by
unc ion con e ges
o
a
ixed
poin
(y*)
o
o
a
limi
cycle
o
a
pe iod
p
which
con ains
a
y*
poin ,
hen
i
is
easy
o
p o e
ha
Lyapuno
exponen
A
<
0.
When
Lyapuno
exponen
Z
0
he
ini ial
pe u ba ion
will
emain
wi h
,
i.e.
ajec o ies
nei he
di e ge
no
con e ge,
hei
ini ial
dis ance
emains
cons an .
This
kind
o
beha io
is
ypical
o
a
cons an
pe iodical
o bi
(Sano
and
Sawada,
1985;
Wol
e
al.,
1985;
McCa ey
e
al.,
1992).
I
he
sys em
is
h ee-dimensional
(i.e.
i
con ains
h ee
s a e
a iables)
he
possible
combina ion
o
signs
and
he
a ac o s
hey
desc ibe
a e:
(+,
0,
-),
o
a
s ange
a ac o ;
(0,
0,-),
a
quasi-pe iodical
a ac o
known
as
o us;
(0,-,
-),
limi
cycle
and
),
a
ixed
poin .
I
in
Eq.
(1)
we
ake
loga i hms
and
we
eplace
d,
by
i s
ma hema ical
exp ession
we
ob ain
A
-ln
In
(2)
Y00
7
I
he
exp ession
(2)
has
a
limi
as
oo
we
de ine
ha
limi
o
be
he
Lyapuno
exponen :
A=
-.oo
lim-llnl
lim
l
ln
(yo
+
do)
(yo)
do
(3)
lim
1
lnl ’(y )l
---,oo
=0
(4)
Equa ion
(4)
ells
us
ha
he
Lyapuno
exponen
is
he
a e age
o
he
na u al
loga i hm
o
he
absolu e
alue
o
he
de i a i es
o
he
disc e e
dynamical
sys em
e alua ed
in
he
ajec o y
poin s
o
ime
se ies
conside ed.
I
he
applica ion
o
he
disc e e
dynamical
sys em
o
wo
close
ajec o ies
e en ually
leads
us
o
sepa a e
poin s,
hen
he
absolu e
alue
o
he
de i a i e
o
is
g ea e
han
1
when
we
e alua ed
a
hose
ajec o y
poin s.
I
he
absolu e
alue
is
g ea e
han
1,
i s
co esponding
loga i hm
is
also
posi i e.
I
he
poin s
o
he
ajec o y
con inue
o
di e ge,
hen
he
a e age
o
he
loga i hms
o
he
de i a i es
is
posi i e.
I
we
calcula e
Lyapuno
exponen
o
a
sample
o
ini ial
poin s
and
we
a e age
he
esul s,
we
can
de ine
he
a e age
Lyapuno
exponen
(X)
o
he
sys em.
An
uni-
dimensional
disc e e
sys em
has
chao ic
ajec o ies,
o
ce ain
pa ame e
alues
on
which
i s
beha io
depend,
i
he
a e age
o
Lyapuno
exponen s
(X)
is
posi i e.
The
exp ession
(4)
o
calcula ing
A
equi es
ha
he
shape
o
he
disc e e
dynamical
sys em
be
known.
Bu ,
wha
happens
when
we
do
no
know
he
sys em
bu
know
one
ime
se ies
o
a
ele an
s a e
a iable?
S udies
conce ning
non-linea
dynamics
ha e
sugges ed
wo
app oxima ions
o
he
es ima ion
o
Lyapuno
exponen :
di ec
me hods
o
di ec
es ima ion
and
Jacobian
me hods
(Guckenheime ,
1982;
Eckmann
and
Ruelle,
1985;
Wol
e
al.,
1985;
McCa ey
e
al.,
1992;
Nychka
e
al.,
1992;
Aba banel
e
al.,
1993;
Damming
and
Mi schke,
1993).
Di ec
me hods
calcula e
Lyapuno
exponen
di ec ly
om
he
ime
se ies,
wi hou
any
addi ional
assump ions
o
app oxima ions
abou
he
subjacen
dynamical
sys em.
Since
he
subjacen
sys em
and
i s
dimensions
a e
unknown,
we
calcula e,
using
he
econs uc ion
ec o
me hod
and
o
di e en
dimensions,
exponen s
un il
ha
ceases
o
a y
signi ican ly
as
he
dimension
o
econs uc ed
space
inc eases.
Al hough
he
disad an age
o
his
p ocedu e
is
ha
i
only
p o ides
he
la ges
Lyapuno
exponen ,
we
will
discuss
his
me hod
below.
LYAPUNOV
EXPONENT
IN
SHORT
TIME
SERIES
43
’J’o
20
FIGURE
Scale
egion
o
a
se ies
o
dis ances
ob ained
om
y +l
4y (1
y ).
Le
us
de ine
yl,Ym,...,Y ,
as
he
elemen s
o
a
ime
se ies
and
dmax
as
he
maximum
dis ance
be ween
wo
poin s
o
he
se ies
equi ed
o
be
conside ed
as
"in ini esimally"
close.
I
he
sys em
beha es
chao ically
o
poin s
Yi
and
yj;
wi h
a
dis ance
do--<
dmax,
he
di e gence
o
close
ajec o ies
will
be
shown
in
he
sequence
di e ences
do
lyj
Yil
d
lYj+
Yi+ll
d2
[Yj+2
Yi+m[
d
ly+
yi+ l
This
will
show
an
exponen ial
inc ease,
o
a
leas
he
mean,
T.
Wi h
his
me hod
o
calcula ing
,
we
ind
wo
close
ajec o ies
in
he
s a e
space
and
we
calcula e
he
se ies
o
dis ances,
which
de i e
om
hese
wo
ini ial
condi ions.
Al hough
in
gene al,
calcula ing
Lyapuno ’s
maximum
exponen
is
easy,
we
belie e
ha
i
is
wo h
conside ing
a
ew
aspec s.
We
assume
a
sepa a ion
a e
o
exponen ial
app oxi-
ma ion
be ween
wo
close
ajec o ies.
Fo
a
gi en
ime
se ies
i
is
necessa y
o
demons a e
his
assump ion.
One
way
o
doing
his
is
by
plo ing
he
na u al
loga i hm
o
he
di e ences
(ln
d )
as
a
unc ion
o
index
T
(see
Fig.
1).
I
he
di e gence
is
exponen ial,
he
poin s
(ln
d ,
T)
will
app oxima e
a
line
gi en
by
he
ollowing
exp ession
in
d
In
do
+
AT
The
slope
o
he
i ed
line--gene ally
by
leas
squa es--is
hen
he
alue
o
Lyapuno
exponen
The
dispe sion
diag am
is
ne e
exac ly
a
line
because
he
exponen ial
di e gence
a ies
along
he
a ac o
and
eaches
a
maximum
when
i
is
compa able
o
he
a ac o ’s
diame e ,
which
is
de ined
as
he
maximum
dis ance
be ween
he
poin s
o
he
ajec o y
e ol ed
o e
he
a ac o .
Two
egions
a e
usually
dis inguished:
one
ha
is
called
scale
egion
o
which
he
line
is
i ed
and
ano he
one
in
which
ln
d
emains
mo e
o
less
cons an
as
T
inc eases.
Adjus ing
a
line
wi h
leas
squa es
o
he
scale
egion
p o ides
he
measu emen
o
he
dominan
Lyapuno
exponen
and
he
accu acy
o
he
adjus men
On
he
o he
hand,
he
alues
o
)
can,
and
gene ally
depend
on
Yi
alues
chosen
o
be
he
ini ial
condi ions,
o
a he ,
he
alues
o
he
ini ial
dis ance
(Id01)
be ween
hem.
To
cha ac e ize
he
subjacen
a ac o
o
a
gi en
ime
se ies
we
ha e
o
calcula e
he
a e age
co esponding
alue
o
he
se
o
Lyapuno
exponen s
ob ained
om
a
numbe
o
ajec o ies
which
ollow
he
condi ion
(5).
do
lyi
yjl
<-
dmax
A
a
p ac ical
le el,
se e al
ques ions
a ise
conce ning
he
ime
span
equi ed
be ween
poin s
Yi
and
yj
so
ha
hey
can
be
conside ed
as
ini ial
condi ions
o
wo
ajec o ies,
he
leng h
o
he
se ies,
he
numbe
o
ini ial
condi ions
o
dis ances,
he
numbe
o
i e a ions
o
op imal
e olu ion
ime
equi ed
and
he
ini ial
dis ance
in
o de
o
conside
poin s
as
in ini esimally
close
in
he
phase
space.
We
do
no
ha e
many
answe s
o
he
ma e s
men ioned
in
he
las
pa ag aph.
Howe e ,
we
ha e
ound
some
sugges ions,
which
ha e
esul ed
om
some
simula ion
expe imen s
made
wi h
speci ic
dynamical
sys ems
in
which
he
heo e ical
alue
o
Lyapuno ’s
maximum
exponen
is
known,
in
his
s udy,
we
ha e
ga he ed
some
o
he
mos
gene al
sugges ions
made
in
he
esea ch
ma e ial
ha
we
ha e
e ised.
The
ini ial
sepa a ion
equi ed
be ween
wo
poin s
so
ha
hese
can
be
conside ed
as
ini ial
condi ions
o
wo
di e en
ajec o ies
in
he
econs uc ed
phase
space
ends
o
be
ela ed
o
wha
is
called
o bi al
pe iod
(Wol
e
al.,
1985;
Theile ,
1986).
This
is
he
ime,
which
a
sys em
akes
o
co e
an
o bi .
I
is
ecommended
ha
he
ini ial
sepa a ion
be ween
wo
poin s
should
be
a
leas
one
o bi al
pe iod.
The
di icul y
o
applying
his
ecommenda ion
lies
in
ha
he
shape
o
he
dynamical
sys em
gene a ing
he
da a
mus
be
known
since
i
is
on
sys em
i sel
ha
he
calcula ion
o
he
o bi al
pe iod
is
based.
In
he
cases
o
which
he
shape
o
he
dynamical
sys em
is
unknown,
we
can
ollow
he
ecommenda ions
gi en
by
Hilbo n
(1994)
and
Theile
(1986).
They
main ain
ha
he
ini ial
sepa a ion
equi ed
be ween
wo
poin s
so
ha
hey
can
be
conside ed
as
ini ial
condi ions
o
wo
di e en
ajec o ies,
mus
be
g ea e
han
wha
is
known
as
au o-co ela ion
ime
(-)
which
is
gi en
by
he
exp ession
(6).
1
-
(6)
ln(1/p)
In
Eq.
(6)
p
is
he
au oco ela ion
coe icien
o
lag
1.
When
p
app oaches
1,
Eq.
(6)
changes
and
becomes
Eq.
(7).
1
-
(7)
As
a
as
he
numbe
o
ini ial
condi ions
equi ed
is
conce ned,
we
ha e ound
se e al
ecommenda ions.
Sano
44
A.M.L.
JIMINEZ
e
al.
o.
8i
o
o
4
o
Io
20
30
4O
so
FIGURE
2
E olu ion
o
he
dis ances
be ween
wo
se ies
gene a ed
om
he
logis ic
map
o
a
4
and
do
0.001.
and
Sawada
(1985)
es ablish
a
lowe
limi
which
will
depend
on
he
dimension
o
he
econs uc ed
space;
o
hem,
he
numbe
o
ini ial
condi ions
equi ed
o
he
es ima ion
o
Lyapuno ’s
maximum
exponen
mus
be
highe
o
equal
o
he
dimension
o
he
econs uc ed
space
(N
>--
de).
Hilbo n
(1994)
ecommends
om
30
o
40
ini ial
condi ions
dis ibu ed
o e
he
a ac o .
O he
au ho s
es ablish
a
dependence
on
he
dimension
o
he
econs uc ed
phase
space
and
on
he
ini ial
ole ance
in
o de
o
conside
poin s
as
in ini esimally
close.
The
exp ession
(8),
by
Dimmig
and
Mi schke
(1993),
p o ides
he
numbe
o
ini ial
condi ions
equi ed
o
cha ac e iz-
ing
he
a ac o
acco ding
o
Lyapuno ’s
maximum
exponen
N
dmax/
(8)
whe e
d,
is
he
dimension
o
he
econs uc ed
space
and
alma
is
he
maximum
ini ial
dis ance
equi ed
o
conside
wo
ajec o ies
as
close.
Wol
e
al.
(1985)
ecommend
om
l0
de
o
30
de
Fo
each
o
he
se ies
o
dis ances
we
calcula e
he
local
Lyapuno
exponen
(Ado).
The
a e age
Lyapuno
exponen
()
is
gi en
by
(9)
whe e
N
is
he
numbe
o
se ies
o
dis ances
conside ed.
Ano he
aspec
o
ake
in o
accoun
is
he
numbe
o
i e a ions
T,
op imal
e olu ion
ime
o
leng h
o
he
se ies
o
dis ance
ha
a e
sui able
o
calcula ing
he
Lyapuno
exponen .
In
he
case
o
chao ic
sys ems,
we
know
ha
con e gence
o
heo e ic
alues
depends,
among
o he s
a iables,
on
he
use
o
an
e olu ion
ime
(T)
ha
is
no
oo
la ge.
Thus,
he
exponen
b ings
he
sensi i i y
o
he
ini ial
condi ions
and
no
he
con e gence
p oduced
by
he
bounda y
o
he
a ac o
alues
o
a
egion
o
he
phases
space
and
o
he
ini ial
dis ance
(do)
o
conside
wo
neighbo ing
ajec o ies.
We
know
ha
he e
is
di e gence
in
chao ic
dynamics
bu
a
he
same
ime,
due
o
he
olding
mechanism,
he
alues
go
h ough
poin s
ha
a e
in ini esimally
close
o
p e ious
alues.
In
Fig.
2
we
ha e
shown
he
e olu ion
o
he
dis ances
be ween
wo
se ies
gene a ed
om
he
logis ic
map,
o
a
4
and
o
he
ini ial
condi ions
y01
0.1
and
y02
0.101.
A
la ge
T
alue
may
p oduce
an
unde es ima ion
o
he
Lyapuno
exponen .
Th ee
c i e ia
ha e
been
p oposed
o
ix
he
numbe
o
i e a ions
o
e olu ion
imes
(T):
(a)
To
es ablish,
a
p io i,
a
ixed
e olu ion
ime
(i.e.
T
10),
(b)
To
es ablish
a
inal
dis ance
ha
can
be
when
he
a ac o
diame e
o
a
pe cen age
is
eached.
In
his
case,
he
ime
o
e olu ion
would
be
a iable
and
(c)
To
iden i y
in
he
g aphic
In
d
be o e
T
in
he
scale
egion.
Finally,
he
dis ance
dmax,
o
he
limi ed
sys ems
ha
in e es
us,
canno
be
oo
la ge.
Due
o
he
ac ha
he
alues
Yi
a e
cons ained
in
size,
he
ini ial
dis ances
canno
be
la ge
han
he
di e ence
be ween
he
maximum
alue,
Ymix,
and
he
minimum
alue,
Ymin.
Mo eo e ,
he e
a e
p ac ical
limi s
when
de e mining
he
ini ial
dis ance
o
he
ini e
p ecision
o
he
da a.
The
numbe
o
decimals
is
an
in e io
limi
o
he
ini ial
dis ance.
Fo
example,
i
he
da a
is
egis e ed
wi h
h ee
decimals,
i
would
be
senseless
o
ques ion
a
di e ence
lesse
han
0.001.
Ano he
e ec
o
he
ini e
p ecision
is
ha
we
can
encoun e
epea ed
da a.
To
summa ize,
he
li e a u e
abou
non-linea
dynamics
only
o e s
a
ew
ecommenda ions,
which
some imes
a e
di e gen ,
abou
he
c i e ia
o
be
used
in
o de
o
selec
he
op imal
calculus
pa ame e s
in
he
es ima ion
o
Lyapuno
exponen s
by
di ec
me hods.
These
ew
ecommenda ions
a e
ci cumsc ibed
o
he
analysis
o
chao ic
sys ems.
We
ha e
ound
no
ecommenda ion
o
he
es ima ion
o
A
s a ing
om
he
ime
se ies
o
classic
sys ems.
The
eason
o
his
is
he
in e es
in
dis inguishing
a iabili y
due
o
a
chao ic
beha io
o
de e minis
dynamic
sys ems
o
a iabili y
caused
by
whi e
noise
o
linea
s ochas ic
p ocesses,
and
less
in
he
iden i ica ion
o
non-
linea
e ms
om
he
analysis
o
ime
se ies.
In
his
s udy,
we
ha e
cen e ed
in
he
dependence
o
he
Lyapuno
exponen ,
ob ained
by
means
o
di ec
es ima ion,
o
he
ini ial
dis ance
and
he
ime
e olu ion.
We
ha e
used
gene a ed
se ies
o
chao ic
sys ems
and
gene a ed
se ies
o
classic
sys ems
wi h
a ying
complex-
i y.
To
gene a e
he
se ies
we
ha e
used
he
logis ic
map
(10).
Y +l
ay (1
Y )
(10)
We
know
ha
he
logis ic
equa ion
is
a
s uc u ally
uns able
sys em.
Tha
is,
i s
beha io
depends
on
he
alue
o
he
pa ame e
a.
In
Table
I
we
ha e
speci ied
he
alues
o
a
used
in
his
esea ch,
i s
beha io
and
he
diame e
o
he
co esponding
a ac o
(Hilbo n,
1994;
S oga z,
1994).
METHOD
In
his
sec ion,
we
desc ibe
he
dependence
o
he
Lyapuno
exponen
bo h
on
he
ini ial
dis ance
and
he
LYAPUNOV
EXPONENT
IN
SHORT
TIME
SERIES
45
TABLE
Value
o
a
Beha io
Diame e
o
a ac o
Lyapuno
exponen
3.2
Limi
cycle
(p
2)
3.52
Limi
cycle
(p
2)
3.55
Limi
cycle
(p
8)
3.58
Chaos
(low
ex en )
4
Chaos
(high
ex en )
0.2910.51,0.81]
-0.91
0.5110.37,0.88]
-0.19
0.53[0.36,0.89]
-0.1
0.56[0.34,0.90]
0.1
1[0,1]
0.69
e olu ion
ime
in
se ies
gene a ed
by
he
logis ic
map.
We
w o e
a
p og am
in
he
Ma hema ica
p og amming
language
( .
2.1
o
Windows)
and
we
gene a ed
se ies
wi h
close
ini ial
alues.
The
p oximi y
c i e ia
used
was
he
one
ecommended
by
Sano
and
Sawada
(1985)
acco ding
o
which
wo
gene a ed
se ies
o
a
dynamic
sys em
a e
conside ed
o
be
close
i
he
ini ial
dis ance
be ween
hem
(do)
is
be ween
1
and
5%
o
he
a ac o
diame e .
Wi h
he
p og am
made
o
gene a e
he
da a
and
gi en
ha
sensi i i y
o
insensi i i y
o
ini ial
condi ions
is
a
cha ac e is ic
o
a
g oup
o
ajec o ies
wi h
close
ini ial
condi ions,
o
he
alues
a
3.2,
3.52,
3.55,
3.58,
4}
We
ini ially
gene a ed
50
se ies
o
N--
100
da a
wi h
ini ial
condi ions
(Yo)
whose
dis ance
was
1%
o
he
a ac o
diame e
(see
Table
I).
F om
he
50
se ies
gene a ed
we
ob ained:
49
se ies
o
dis ances
wi h
do
1%,
48
wi h
do
2%,
47
wi h
do
3%,
46
wi h
do
4%
and
45
wi h
do
5%.
The
ini ial
alue
o
he
i s
se ies
gene a ed
(Yo)
coincided
wi h
he
minimum
alue
o
he
a ac o
ampli ude
(see
Table
I)
wi h
he
excep ion
o
he
se ies
gene a ed
o
a
4,
whe e
we
s a ed
om
y0--
0.1.
Fo
his
las
alue,
we
used
h ee
addi ional
dis ances
among
he
ini ial
condi ions
0.1,
0.25
and
0.5%
and
50
dis ance
se ies
we e
gene a ed.
The
Lyapuno
exponen
o
1
<--
T
--<
99
and
he
ini ial
dis ance
(do)
we e
ob ained
by
means
o
he
exp ession
A=ln
In
(11)
Yio
YjO
-o
We
calcula ed
he
a e age
Lyapuno
exponen
(X)
as
es ima o
o
de
A
o
he
coe icien s
ob ained
wi h
exp ession
(11)
o
each
do
and
T.
We
g aphically
ep esen ed
X
as
a
unc ion
o
index
T
o
each
alue
o
do
in
o de
o
s udy
he
incidence
o
hese
pa ame e s.
RESULTS
We
o ganized
he
esul s
in
wo
di e en
sec ions.
In
he
i s
pa ,
we
show
hose
esul s
co esponding
o
he
T
T
-,2
e)e
3.2,
do=5
iles
0,04
FIGURE
3
Beha io
o
X
acco ding
o
T
when
inc easing
he
do
o
a
3.2.

46
A.M.L.
JIMNEZ
e
al.
FIGURE
4
Beha io
o
X
when
inc easing
do
o
10
and
20%.
alues
o
a
whose
beha io
con e ges
on
a
classic
a ac o .
In
he
second
pa ,
we
o e
he
esul s
o
he
alues
o
a
wi h
a
chao ic
beha io .
We
ha e
p e e ed
o
show
he
g aphics
co esponding
o
he
e olu ion
o
X
acing
T
ins ead
o
he
alue
ables
because
we
see
hem
as
mo e
illus a i e
and
easie
o
in e p e .
Classic
A ac o
a
3.2
In
Fig.
3,
we
ep esen ed
he
a e age
Lyapuno
exponen
(X)
as
a
unc ion
o
he
e olu ion
ime
(T)
o
he
se
o
dis ances
ha
we e
conside ed.
In
he
di e en
g aphics
o
Fig.
3,
we
ha e
d awn
a
b oken
line
h ough
he
alue
o
he
heo e ic
Lyapuno
exponen .
The
con e gence
o
his
alue
can
be
seen
wi h
he
inc ease
o
he
e olu ion
ime
T.
The
con e gence
o m
is
independen
om
he
ini ial
dis ance
(do)
conside ed.
In
all
cases
he e
is
an
oscilla o y
dec ease
o
X.
Fo
he
se
do
{1%,2%,3%}
he
con e gence
s ops
in
T
14,
ob aining
he
alue
o
X
closes
o
he
heo e ic
when
T
13.
-,z,
-
ba=3.52,
d=4%,
Bias=O.01,
T,,=17
,2
0,0
-,2
T
c)==3.SZ
=
=0.00
’.=7
d)
3.52,
d
4%,
T
e)
3.52,
d=
5,
Bias
0.05,
T,,
17
FIGURE
5
Beha io
o
X
acco ding
o
he
e olu ion
ime
(T)
o
a
3.52.
LYAPUNOV
EXPONENT
IN
SHORT
TIME
SERIES
47
T
b)
3,55,
do
2%,
8as
O.01,
T
c)
3,55,
d
3%,
Odas
<0.0I
T
d)
SS,
d,=,m,
nab
002
Bias
0,01,
T.-7
FIGURE
6
Beha io
o
X
acco ding
o
he
e olu ion
ime
(T)
o
a
3.55.
Fo
he
se
do
{4%,
5%
he
con e gence
s ops
in
T
16.
The
closes
mean
Lyapuno
exponen
(A)
is
ob ained
o
T
15
and
T
13
( he
bias
is
0.04).
To
see
i
inc easing
he
ini ial
dis ance
would
dec ease
he
bias
(X-A),
we
analyzed
in
16
and
15
se ies
o
dis ances
he
beha io
o
A
acco ding
o
he
e olu ion
ime
(T)
when
inc easing
he
ini ial
dis ance
(do)
be ween
he
ajec o ies
o
10
and
20%
o
he
a ac o
diame e .
We
ha e
p esen ed
he
esul s
in
Fig.
4.
I
can
be
obse ed
(g aphic
(a)),
how
he
alue
o
he
bias
and
he
shape
o
he
con e gence
a e
simila
o
hose
ob ained
o
he
ini ial
dis ances
analyzed
p e iously.
On
he
con a y,
when
he
alue
o
he
ini ial
dis ance
inc eases
o
20%
he
bias
inc eases
d ama ically
o
0.51.
a
3.52
Figu e
5
shows
he
beha io
o
A
acco ding
o
he
e olu ion
ime
o
he
uni
o
ini ial
dis ances
conside ed:
do
{1%,2%,3%,4%,5%}
As
wi h
he
se ies
ep esen ed
in
Fig.
3,
o
a
3.52
he
mean
Lyapuno
exponen
declines
oscilla o y
when
T
inc eases.
Mo eo e ,
some
di e ences
can
be
obse ed
in
ela ion
o
do.
In
he
g aphics
o
Fig.
5,
we
ha e
d awn
a
con inuous
line.
This
is
pe pendicula
o
he
axis
o
he
in e sec ion
o
he
alue
o
T,
which
p o ides
he
bes
es ima ion
o
A.
As
is
cus oma y,
a
b oken
line
ep esen s
he
heo e ic
alue
o
he
Lyapuno
exponen .
We
can
see
(g aphics
a,
b
and
c
in
Fig.
5)
ha
X
ends
o
he
heo e ic
alue
when
he
ini ial
dis ances
a e
1-3%.
When
inc easing
he
ini ial
dis ance
o
4
and
5%
(g aphics
d
and
e
in
Fig.
5),
he
limi
o
X
is
no
he
heo e ic
alue
bu
a
la ge
one.
Fo
he
alues
o
dis ances
and
e olu ion
imes
conside ed,
he
di ec
es ima ion
p o ides
alues
ha
a e
biased
posi i ely
o
A.
a
3.55
Figu e
6
shows
he
beha io
o
X
acco ding
o
T
o
he
dis ances
conside ed.
In
he
g aphics
in
Fig.
6,
we
ha e
d awn
an
o dina e
line
A--0
( he
g ay
line)
wi h
he
aim
o
assessing
possible
quali a i e
e o s
when
iden i ying
he
subjacen
a ac o
o
he
da a
gene a ing
sys em.
The
To
(op imal)
is
he
alues
o
T
o
which
he
bias
is
less.
In
g aphs
(a)
and
( )
in
Fig.
6,
we
can
see
an
ini ial
pe iod
wi h
g ea
a iabili y
in
which
A
oscilla es
be ween
48
A.M.L.
JIMINEZ
e
al.
T
3.56,
dn
1%
slow
and
p ac ically
s abilizes
i sel
oscilla ing
be ween
-0.04
and
-0.02
when
he
ini ial
dis ance
is
1%,
as
can
be
seen
in
Fig.
7.
Fo
he
es
o
he
do
wi h
he
e olu ion
ime,
oscilla ion
s abilizes
be ween
0.03
and
0.001.
In
any
case,
we
can
deduce
om
he
Fig.
6,
ha
using
sho
and
e en
e olu ion
imes
( i s
sec ion
o
he
dispe sion
diag ams)
can
be
mo e
p oblema ic
han
using
longe
e olu ion
imes
as,
al hough
he
bias
inc eases
wi h
T,
he e
a e
no
quali a i e
e o s
no ed.
FIGURE
7
Oscilla ions
o
X
in
he
in e al:
T
[60,
100].
he
zone
o
chao ic
beha io
(A
>
0)
o
he
e en
T
alues,
and
he
zone
o
ecu en
beha io
(X
>
0)
o
he
odd
alues.
A e
his
i s
span,
when
T
g ows
o
he
alue
o
A
ha
is
highe
han
he
heo e ical
alue
(X
-0.1),
he
di ec
es ima ions
o
con e ge
slowly
and
in
an
oscilla ing
way.
The
inc ease
in
he
ini ial
dis ance
p ocess
b ings
he
bounda y
o
he
con e gence
p ocess
o
X
o
0.
In
he
in e al
5
-<
T
--<
11
we
can
ind
he
alues
ha
p o ide
he
bes
es ima ions
o
.
A
common
alue
o
To
ha
p o ides
eliable
es ima ions
(bias
-<
0.02)
o
he
se
o
dis ances
unde
conside a ion
is
To
7.
F om
his
ini ial
pe iod,
bo h
he
decline
o
,(
o
e en
alues
o
T,
as
well
as
he
g ow h
o
odd
alues,
is
e y
Chao ic
Beha io
a
3.58
Amongs
he
many
alues
o
a
whose
beha io
is
chao ic
in
he
in e al
[3.58,
4],
we
used
p ecisely
he
ex emes,
which
co espond
o
he
s ange
a ac o
o
he
smalles
and
la ges
diame e s,
espec i ely.
Figu e
8
shows
he
esul s
ela i e o
he
o m
o
he
dependence
o
X
as
opposed
o
he
e olu ion
ime.
In
he
se
o
g aphs
in
Fig.
8,
we
can
dis inguish
h ee
sec ions:
in
he
i s
sec ion
o
app oxima ely
1
<_
T
<--
4,
we
can
see
a
sha p
inc ease
o
X
owa ds
he
heo e ical
alue.
T
0
20
40
0
80
00
b)a=3.S&
do=2%
To={4,8}.Bla$=O
.58.
=3
%,
T,.,
14,
8)
I
0.01
o )a=3.58,
do=4%,
To=
[4.
8),
Bi=
e)==3.,.’,8,
do=5%.
T.,={4.S}.
IIZI
.c0.01
do
T
7,00
8,00
FIGURE
8
Beha io
o
X
acco ding
o
he
e olu ion
ime
(T)
when
inc easing
he
ini ial
dis ance
o
a
3.58.
LYAPUNOV
EXPONENT
IN
SHORT
TIME
SERIES
49
,10
,05
,05
.04
.02
.01,
-.01
-,0
-,0
T
c)
3.58
cJo
-.3%
,05
,03
01o
-.02
-,03
-,04
"
-,0
d)a
3.5
do
4%
,04
,02
,01,
,00
-,01
-,0.
-,0
-,04
,0
::N
FIGURE
9
Beha io
o
X
o
T
>
8
o
a
3.58.
We
can
conside
he
second
sec ion
as
s abiliza ion,
in
which
A
oscilla es
a ound
a
cons an
alue
e y
nea
o
he
heo e ical
alue
o
he
e en
alues
o
T.
This
second
span
becomes
p og essi ely
sho e
as
he
ini ial
dis ances
g ow.
Thus,
o
do--1%,
he
alues
o
X
emain
p ac ically
cons an
un il
T
25,
whils
o
do
5%,
his
sec ion
inishes
in
T
8.
Du ing
his
ime
o
s abiliza ion,
as
he
ini ial
dis ance
inc eases,
e o
is
mo e
likely
in
iden i ying
he
ype
o
a ac o
p esen
in
he
da a,
especially
o
he
odd
alues
o
T.
The
hi d
sec ion,
which
is
clea ly
iden i iable
in
he
igu es,
is
he
longes .
In
his
sec ion,
A
oscilla es
wi h
a ia ions
ha
a e
p ac ically
cons an
in
size,
a ound
0,
independen
o
he
ini ial
dis ance.
The
inc ease
in
e olu ion
ime
o
highe
han
op imal
T
unde es ima es
he
alue
o
X
I
seems
ha
wi h
T,
he
a e age
exponen
X
would
be
cen e ed
on
0
o
all
he
ini ial
dis ances.
We
ha e
widened
his
sec ion
speci ically
in
o de
o
obse e
whe he
he
beha io
in
his
a ea
is
eally
independen
o
he
ini ial
dis ance
o
no .
The
esul s
a e
shown
in
Fig.
9.
Taking
he
alues
o
X
o
T
>
8,
we
can
see
ha ,
wi h
he
ini ial
dis ances
do
inc eases
he
possibili y
o
quali a i e
e o
when
using
longe
e olu ion
imes.
In
g aph
(a)
in
Fig.
9,
p ac ically
100%
o
he
A
alues
a e
g ea e
han
0.
He e,
al hough
he
bias
is
sha p,
ne e heless
he
quali a i e
conclusion
ega ding
he
na u e
o
he
a ac o
con ained
in
he
se ies
would
be
co ec .
G aphs
(b)-(d)
in
Fig.
9
shows
how
he
dis ibu ion
o
alues
a ound
A--0
in e s
wi h
he
inc ease
in
he
ini ial
dis ance
and
he
numbe
o
nega i e
exponen s
inc eases
p og essi ely.
The
beha io
o
X
in
ela ion
o
T
is
simila
o
ha
obse ed
o
he
alues
o
a
wi h
ecu en
beha io
a
in e als
o
2,
4
and
8.
In
g aph
( )
in
Fig.
8
we
can
see
how,
o
he
alues
o
T
ha
cons i u e
he
i s
and
second
sec ion
o
he
e olu ion
o
X
o
he
se
o
ini ial
dis ances,
he
bias
is
independen
o
hese.
a--4
G aphs
(a)
and
(h)
in
Fig.
10
show
he
dependence
o
X
wi h
espec
o
he
e olu ion
ime
o
he
se
o
do
ha
was
applied.
In
g aphs
(b)-(g)
(see
Fig.
10)
we
can
app ecia e
a
e y
apid
ini ial
inc ease
which
is
mo e
o
less
lineal,
owa ds
he
heo e ical
alue
o
X.
This
g ow h
is
in e up ed
ab up ly
in
A
alues
below
he
heo e ical
alue.
Fo
g owing
alues
o
he
se
o
ini ial
dis ances
do--=
1%,
2%,
3%,
4%,
5%
},
he
maximum
X
alues
eached
a e:
0.59, 0.58,
0.55,
0.55
and
0.53.
I
appea s
ha
we
can
con i m
a
di ec
ela ionship
be ween
he
bias